Parametric Schur convexity and arrangement monotonicity properties of partial sums (Q1893367)

Let \(X_i (\theta_i)\), \(i = 1,2, \ldots, n\), be independent random variables such that the distribution of \(X_i (\theta_i)\) is determined by the value of \(\theta_i\). Denote \(S (\theta) = (X_1 (\theta_1)\), \(X_1 (\theta_1) + X_2 (\theta_2), \ldots, \sum^n_{i = 1} X_i (\theta_i) )\). The authors give sufficient conditions on \(f : R^n \to R\) and on \(\{X_i (\theta_i)\), \(\theta \in \Theta\}\) under which \(f(S (\theta))\) have some stochastic arrangement monotonicity and stochastic Schur convexity properties. The next result is typical. Suppose that \(\{X_i (\theta), \theta \in \Theta\}\) is stochastically increasing and convex in the sense of the hazard rate ordering. Then we have that \(\sum^k_{i = 1} X_i (\theta_i)\), \(k = 1,2, \ldots,n\), is stochastically increasing and Schur-convex in \(\theta \in {\mathcal X}= \{\theta \in \Theta^n : \theta_1 \geq \theta_2 \geq \cdots \geq \theta_n\}\) in the sense of the usual stochastic ordering.